On the linearization of operators related to the full waveform inversion in seismology

Kirsch, Andreas; Rieder, Andreas

doi:10.1002/mma.3037

Cited by 22 publications

(25 citation statements)

References 15 publications

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“…In contrast to the publications mentioned before, we consider the identification of stored energy functions, which are spatially variable, from time‐dependent boundary data. The article by Kirsch and Rieder can be seen in some sense as an analogon for the acoustic wave equation.…”

Section: Introductionmentioning

confidence: 99%

On the linearization of identifying the stored energy function of a hyperelastic material from full knowledge of the displacement field

Seydel

Schuster

2016

Math Methods in App Sciences

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We compute a local linearization for the nonlinear, inverse problem of identifying the stored energy function of a hyperelastic material from the full knowledge of the displacement field. The displacement field is described as a solution of the nonlinear, dynamic, elastic wave equation, where the first Piola–Kirchhoff stress tensor is given as the gradient of the stored energy function. We assume that we have a dictionary at hand such that the energy function is given as a conic combination of the dictionary's elements. In that sense, the mathematical model of the direct problem is the nonlinear operator that maps the vector of expansion coefficients to the solution of the hyperelastic wave equation. In this article, we summarize some continuity results for this operator and deduce its Fréchet derivative as well as the adjoint of this derivative. Because the stored energy function encodes mechanical properties of the underlying, hyperelastic material, the considered inverse problem is of highest interest for structural health monitoring systems where defects are detected from boundary measurements of the displacement field. For solving the inverse problem iteratively by the Landweber method or Newton‐type methods, the knowledge of the Fréchet derivative and its adjoint is of utmost importance. Copyright © 2016 John Wiley & Sons, Ltd.

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Section: Introductionmentioning

confidence: 99%

On the linearization of identifying the stored energy function of a hyperelastic material from full knowledge of the displacement field

Seydel

Schuster

2016

Math Methods in App Sciences

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“…In general, the auxiliary problems take the same form as the original problem, however they are inhomogeneous. They contain volume and/or interface (boundary) sources for penetrable (impenetrable) obstacles, all defined in terms of the solution to the original problem and its (possibly higher order) traces 10 . Difficulty arises under low regularity of interfaces/boundaries, since interface (boundary) sources might not have well-defined traces in the canonical boundary spaces H s (Γ), |s| ≤ 1.…”

Section: Auxiliary Problem For Partial Derivatives With Lamé Parametersmentioning

confidence: 99%

“…Figure 1(a). For numerical evaluation using volumed-based discretization, the infinite domain is truncated using absorbing boundary condition (ABC), and the problem we focus on, denoted by OP, is defined on a finite convex domain Ω finite , scattering with penetrable obstacles 1 , to be distinguished with elastic problems on bounded domains such as [8] for elastostatics, [9] in time-harmonic elastodynamics, or [10,11] in elastodynamics.…”

Section: Introductionmentioning

confidence: 99%

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Characterization of partial derivatives with respect to material parameters in a fluid–solid interaction problem

Azpiroz

Barucq

Djellouli

et al. 2018

Journal of Mathematical Analysis and Applications

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For a fluid-solid interaction problem with Lipschitz interface, we investigate the partial Fréchet differentiability of the solutions and the approximate far-fieldpattern with respect to solid material parameters. Differentiability is shown in standard Sobolev framework, and the derivatives are characterized as solutions to inhomogeneous fluid-solid transmission problems. To validate the accuracy of the characterization, we compare analytical values with numerical ones given by Interior Penalty Discontinuous Galerkin (IPDG) in a setting with circular obstacles. Our comparisons also show that IPDG gives results with high precision and incurs almost no effect of discretization error accumulation.

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“…They set up an inexact Newton iteration to this end validating the Fréchet differentiability of the parameter-to-solution map in the spirit of [12]. Boehm and Ulbrich [2] attack the same problem with a semismooth Newton iteration also providing an expression for the Fréchet derivative.…”

Section: Introductionmentioning

confidence: 99%

Inverse problems for abstract evolution equations with applications in electrodynamics and elasticity

Kirsch¹,

Rieder²

2016

Inverse Problems

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Abstract. It is common knowledge -mainly based on experience -that parameter identification problems in partial differential equations are ill-posed. Yet, a mathematical sound argumentation is missing, except for some special cases. We present a general theory for inverse problems related to abstract evolution equations which explains not only their local ill-posedness but also provides the Fréchet derivative of the corresponding parameter-to-solution map which is needed, e.g., in Newton-like solvers. Our abstract results are applied to inverse problems related to the following first order hyperbolic systems: Maxwell's equation (electromagnetic scattering in conducting media) and elastic wave equation (seismic imaging).

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On the linearization of operators related to the full waveform inversion in seismology

Cited by 22 publications

References 15 publications

On the linearization of identifying the stored energy function of a hyperelastic material from full knowledge of the displacement field

On the linearization of identifying the stored energy function of a hyperelastic material from full knowledge of the displacement field

Characterization of partial derivatives with respect to material parameters in a fluid–solid interaction problem

Inverse problems for abstract evolution equations with applications in electrodynamics and elasticity

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